Factoring and Prime factors

Factoring can be described as the process of breaking down a large number into a product of smaller numbers. Those numbers are called factors and the process is called factoring. The process that is the opposite of factoring is called distribution. Some numbers can be factorized in two or more ways. For example, the number 18 can be factorized in the following ways: , or .
Usually, we will need to find the prime factors of a number. Prime factors are the factors of the given number that are also prime numbers. For example, the prime factors of number 12 are the numbers 2, 2, and 3 or .
In the next example, we will use a simple method to factorize the number 90.

-> First, we try to divide a given number by the lowest prime number. If the number is not divisible, we can try to divide it by the next larger prime number. In our case, the first divisor of the number 90 is the number 2. The number 2 is the first prime factor.
-> We will try to divide the result of the previous division by the number 2. We can’t divide 45 by 2, so we move on the next larger prime number, the number 3. The number 45 can be divided by the number 3, and the result of the division is the number 15. The number 3 is the second prime factor.
-> The number 15 can be divided by the number 3 and the result is 5. The number 5 is the third prime number. Note that in this step we didn’t try to divide the number by 2 because we already excluded the number 2 from the possible prime factors when we moved to dividing by the number 3.
-> The number 5 can’t be divided by 3, but it can be divided by the next bigger prime factor, the number 5. The result is 1. The number 5 is the fourth prime number.

In the following example we will find the prime factors of number 210 using the method described above.

-> 210 can be divided by 2 and the result is 105. The number 2 is the first prime number.
-> 105 can’t be divided by 2, so we try the next larger prime number. 105 can be divided by 3 and the result is 35. The number 3 is the second prime number.
-> 35 can’t be divided by 3, but it can be divided by the number 5 and the result is 7. The number 5 is the third prime number.
-> 7 can’t be divided by 5. 7 divided by 7 is 1. The number 7 is the last prime number. -> The prime factors of 210 are .

The concept of factoring is very important because of the other techniques we are going to learn about. Factoring is used in finding the greatest common factor (GCF) and least common multiple (LCM).
The greatest common factor is the largest common number that can divide two numbers. For example, the greatest common factor of 12 and 16 is 4, because it is the largest number that can divide both those two numbers.
The least common multiple of two numbers is the lowest number that can be divided by those two numbers. For example, the least common multiple of 12 and 16 is the number 48, because it is the lowest number that can be divided by both 12 and 16.